Bifurcation theory of attractors and minimal sets in d-concave nonautonomous scalar ordinary differential equations

The paper proves a double saddle-node global bifurcation for scalar d-concave, coercive skew-product flows: if three minimal sets exist at some λ0, then on an open interval (λ−, λ+) there are exactly three hyperbolic copies of the base ordered αλ < κλ < βλ with κλ strictly decreasing; at λ± two copies collide on a σ-invariant residual set to form a nonhyperbolic minimal set; outside [λ−, λ+] a unique attractive hyperbolic copy remains, and the attractor bifurcates discontinuously. These claims are stated and proved in Theorem 5.10 and its proof and corollaries (e.g., nonhyperbolicity and “pinching” at λ±) in the PDF the user supplied . The candidate solution reproduces the same scenario and main conclusions, using somewhat different arguments (e.g., a concave fixed-point count and a graph-transform argument). Its monotonicity in λ matches Theorem 5.5 , and its use of the coercive global attractor interval [αλ, βλ] matches Theorem 5.1 . One substantive issue: the candidate’s Step 1 claims any τλ-minimal set projecting onto Ω is a copy of the base; the paper shows that, at the bifurcation endpoints λ±, the minimal set created by collision is not a copy (it is pinched/nonhyperbolic) . The candidate later acknowledges non-copy pinched sets at λ±, but Step 1 is overstated and needs the qualification “away from the collision parameters” (or “when three minimal sets exist”). Aside from this overreach, the model’s outline is consistent with the paper’s results, and differences are largely methodological: where the paper relies on quantitative (SDC) consequences like Proposition 5.9 and the at-most-three structure/results of Section 4 , the candidate sketches a contraction-of-distances argument via the standardized module bJ,ε (cf. the paper’s definition of (SDC) via bJ,ε) .